Let $A,B$ be $n\times n$ matrices. I am interested in the following ODE in $\mathbb{R}^n$

$$ \frac{dx_t}{dt}=Ax_t+Bx^+_t $$

where $x_t^+=(x^+_{1,t},...,x^+_{n,t})$ and $(\cdot)^+$ is the rectifier: $r^+=max\{0,r\}.$

Does this type of ODE have a name? And are there any known criterias for stability? Has it been studied by anyone in general?

The closest i have found is the "Threshold-Linear networks" studied here for example. I appreciate any reference similar to this system.